Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Analysis of various factors affecting the non-Markovian dynamics associated with a hierarchical environment based on collision model

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

We propose a quantum collision model in which the environment is abstractively divided into two hierarchies including “environment-bus” that has direct interactions with the system and “environment-stations” that has not. Based on the model, we investigate the effects of initial system–environment correlations, initial states of environment, and various interactions on the dynamics of open quantum systems associated genuinely with such a hierarchical environment. We illustrate that the initial quantum correlation between the system and environment leads to a transition from Markovian to non-Markovian dynamics, while for initial classical correlation the transition can only be confirmed to happen when the couplings rather than the correlations in environment are present. In addition, we investigate the degree of non-Markovianity varying with environment initial states and reveal that the interaction strength between two environmental hierarchies plays an important role in it. In particular, we show that in such a hierarchically structured environment the degree of non-Markovianity is not equivalent to memory effects of the environment-stations as a reservoir due to the presence of the environment-bus.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Breuer, H.P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2002)

    MATH  Google Scholar 

  2. Weiss, U.: Quantum Dissipative Systems, 3rd edn. World Scientific, Singapore (2008)

    Book  MATH  Google Scholar 

  3. Rivas, A., Huelga, S.F.: Open Quantum Systems: An Introduction. Springer, Heidelberg (2011)

    MATH  Google Scholar 

  4. Gardiner, C.W., Zoller, P.: Quantum Noise. Springer, Berlin (2002)

    MATH  Google Scholar 

  5. Mogilevtsev, D., Nisovtsev, A.P., Kilin, S., Cavalcanti, S.B., Brandi, H.S., Oliveira, L.E.: Driving-dependent damping of rabi oscillations in two-level semiconductor systems. Phys. Rev. Lett. 100, 017401 (2008)

    Article  ADS  Google Scholar 

  6. Galland, C., Hogele, A., Tureci, H.E., Imamoglu, A.: Non-Markovian decoherence of localized nanotube excitons by acoustic phonons. Phys. Rev. Lett. 101, 067402 (2008)

    Article  ADS  Google Scholar 

  7. Madsen, K.H., Ates, S., Lund-Hansen, T., Loffler, A., Reitzenstein, S., Forchel, A., Lodahl, P.: Observation of non-Markovian dynamics of a single quantum dot in a micropillar cavity. Phys. Rev. Lett. 106, 233601 (2011)

    Article  ADS  Google Scholar 

  8. Tang, J.S., Li, C.F., Li, Y.L., Zou, X.B., Guo, G.C., Breuer, H.P., Laine, E.M., Piilo, J.: Measuring non-Markovianity of processes with controllable system–environment interaction. Europhys. Lett. 97, 10002 (2012)

    Article  ADS  Google Scholar 

  9. Lai, C.W., Maletinsky, P., Badolato, A., Imamoglu, A.: Knight-field-enabled nuclear spin polarization in single quantum dots. Phys. Rev. Lett. 96, 167403 (2006)

    Article  ADS  Google Scholar 

  10. Aharonov, D., Kitaev, A., Preskill, J.: Fault-tolerant quantum computation with long-range correlated noise. Phys. Rev. Lett. 96, 050504 (2006)

    Article  ADS  Google Scholar 

  11. Hu, B.L., Paz, J.P., Zhang, Y.: Quantum Brownian motion in a general environment: exact master equation with nonlocal dissipation and colored noise. Phys. Rev. D 45, 2843 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Strunz, W.T., Yu, T.: Convolutionless Non-Markovian master equations and quantum trajectories: Brownian motion. Phys. Rev. A 69, 052115 (2004)

    Article  ADS  Google Scholar 

  13. Vacchini, B., Breuer, H.P.: Exact master equations for the non-Markovian decay of a qubit. Phys. Rev. A 81, 042103 (2010)

    Article  ADS  Google Scholar 

  14. Zhang, W.M., Lo, P.Y., Xiong, H.N., Tu, M.W.Y., Nori, F.: General non-Markovian dynamics of open quantum systems. Phys. Rev. Lett. 109, 170402 (2012)

    Article  ADS  Google Scholar 

  15. Verstraete, F., Wolf, M.M., Cirac, J.I.: Quantum computation and quantum-state engineering driven by dissipation. Nat. Phys. 5, 633 (2009)

    Article  Google Scholar 

  16. Barnett, S.M., Stenholm, S.: Hazards of reservoir memory. Phys. Rev. A 64, 033808 (2001)

    Article  ADS  Google Scholar 

  17. Zhang, W.M., Feng, D.H., Gilmore, R.: Coherent states: theory and some applications. Rev. Mod. Phys. 62, 867 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  18. Feynman, R.P., Vernon, F.L.: The theory of a general quantum system interacting with a linear dissipative system. Ann. Phys. 24, 118 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  19. Coish, W.A., Loss, D.: Hyperfine interaction in a quantum dot: non-Markovian electron spin dynamics. Phys. Rev. B 70, 195340 (2004)

    Article  ADS  Google Scholar 

  20. Coish, W.A., Fischer, J., Loss, D.: Free-induction decay and envelope modulations in a narrowed nuclear spin bath. Phys. Rev. B 81, 165315 (2010)

    Article  ADS  Google Scholar 

  21. Barnes, E., Cywinski, L., Das Sarma, S.: Nonperturbative master equation solution of central spin dephasing dynamics. Phys. Rev. Lett. 109, 140403 (2012)

    Article  ADS  Google Scholar 

  22. Shabani, A., Lidar, D.A.: Compeletly positive post-markovian master equation via a measurement approach. Phys. Rev. A 71, 020101 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Scarani, V., Ziman, M., Štelmachovič, P., Gisin, N., Bužek, H.: Thermalizing quantum machines: dissipation and entanglement. Phys. Rev. Lett. 88, 097905 (2002)

    Article  ADS  Google Scholar 

  24. Bodor, A., Diósi, L., Kallus, Z., Konrad, T.: Structural features of non-Markovian open quantum systems using quantum chains. Phys. Rev. A 87, 052113 (2013)

    Article  ADS  Google Scholar 

  25. Budini, A.A.: Embedding non-Markovian quantum collisional models into bipartite Markovian dynamic. Phys. Rev. A 88, 032115 (2013)

    Article  ADS  Google Scholar 

  26. Vacchini, B.: Non-Markovian master equations from piecewise dynamics. Phys. Rev. A 87, 030101(R) (2013)

    Article  ADS  Google Scholar 

  27. Gennaro, G., Benenti, G., Palma, G.M.: Entanglement dynamics and relaxation in a few-qubit system interacting with random collisions. Europhys. Lett. 82, 20006 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  28. Low, G.H., Shi, Z.M., Yeo, Y.: Effects of collisional decoherence on multipartite entanglement: how entanglement might not be relatively common. Phys. Rev. A 74, 012307 (2006)

    Article  ADS  Google Scholar 

  29. Gennaro, G., Benenti, G., Palma, G.M.: Relaxation due to random collisions with a many-qudit environment. Phys. Rev. A 79, 022105 (2009)

    Article  ADS  Google Scholar 

  30. Giovannetti, V., Palma, G.M.: Master equations for correlated quantum channels. Phys. Rev. Lett. 108, 040401 (2012)

    Article  ADS  Google Scholar 

  31. Giovannetti, V., Palma, G.M.: Master equation for cascade quantum channels: a collisional approach. J. Phys. B 45, 154003 (2012)

    Article  ADS  Google Scholar 

  32. Rybár, T., Filipov, S.N., Ziman, M., Buzek, V.: Simulation of indivisible qubit channels in collision models. J. Phys. B 45, 154006 (2012)

    Article  ADS  Google Scholar 

  33. Ciccarello, F., Palma, G.M., Giovannetti, V.: Collision-model-based approach to non-Markovian quantum dynamics. Phys. Rev. A 87, 040103(R) (2013)

    Article  ADS  Google Scholar 

  34. Ciccarello, F., Giovannetti, V.: A quantum non-Markovian collision model: incoherent swap case. Phys. Scr. T 153, 014010 (2013)

    Article  ADS  Google Scholar 

  35. Bernardes, N.K., Carvalho, A.R.R., Monken, C.H., Santos, M.F.: Environmental correlations and Markovian to non-Markovian transitions in collisional models. Phys. Rev. A 90, 032111 (2014)

    Article  ADS  Google Scholar 

  36. Ziman, M., Buz̆ek, V.: All (qubit) decoherences: complete characterization and physical implementation. Phys. Rev. A 72, 022110 (2005)

    Article  ADS  Google Scholar 

  37. McCloskey, R., Paternostro, M.: Non-Markovianity and system–environment correlations in a microscopic collision model. Phys. Rev. A 89, 052120 (2014)

    Article  ADS  Google Scholar 

  38. Iles-Smith, J., Lambert, N., Nazir, A.: Environmental dynamics, correlations, and the emergence of noncanonical equilibrium states in open quantum systems. Phys. Rev. A 90, 032114 (2014)

    Article  ADS  Google Scholar 

  39. Garg, A., Onuchic, J.N., Ambegaokar, V.: Effect of friction on electron transfer in biomolecules. J. Chem. Phys. 83, 4491 (1985)

    Article  ADS  Google Scholar 

  40. Thoss, M., Wang, H.B., Miller, W.H.: Self-consistent hybrid approach for complex systems: application to the spin-boson model with Debye spectral density. J. Chem. Phys. 115, 2991 (2001)

    Article  ADS  Google Scholar 

  41. Cao, J.S., Voth, G.A.: A unified framework for quantum activated rate processes. II. The nonadiabatic limit. J. Chem. Phys. 106, 1769 (1997)

    Article  ADS  Google Scholar 

  42. Schönleber, D.W., Croy, A., Eisfeld, A.: Pseudomodes and the corresponding transformation of the temperature-dependent bath correlation function. Phys. Rev. A 91, 052108 (2015)

    Article  ADS  Google Scholar 

  43. Levi, E.K., Irish, E.K., Lovett, B.W.: Coherent exciton dynamics in a dissipative environment maintained by an off-resonant vibrational mode. Phys. Rev. A 93, 042109 (2016)

    Article  ADS  Google Scholar 

  44. Ma, T.T., Chen, Y.S., Chen, T., Hedemann, S.R., Yu, T.: Crossover between non-Markovian and Markovian dynamics induced by a hierarchical environment. Phys. Rev. A 90, 042108 (2014)

    Article  ADS  Google Scholar 

  45. Feng, X.L., Zhang, Z.M., Li, X.D., Gong, S.Q., Xu, Z.Z.: Entangling distant atoms by interference of polarized photons. Phys. Rev. Lett. 90, 217902 (2003)

    Article  ADS  Google Scholar 

  46. Lange, W., Kimble, H.J.: Dynamic generation of maximally entangled photon multiplets by adiabatic passage. Phys. Rev. A 61, 063817 (2000)

    Article  ADS  Google Scholar 

  47. Breuer, H.P., Laine, E.M., Piilo, J.: Measure for the degree of non-Markovian behavior of quantum processes in open systems. Phys. Rev. Lett. 103, 210401 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  48. Laine, E.M., Piilo, J., Breuer, H.P.: Witness for initial system–environment correlations in open-system dynamics. Europhys. Lett. 92, 60010 (2010)

    Article  ADS  Google Scholar 

  49. Dajka, J., Łuczka, J.: Distance growth of quantum states due to initial system–environment correlations. Phys. Rev. A 82, 012341 (2010)

    Article  ADS  Google Scholar 

  50. Wißmann, S., Karlsson, A., Laine, E.M., Piilo, J., Breuer, H.P.: Optimal state pairs for non-Markovian quantum dynamics. Phys. Rev. A 86, 062108 (2012)

    Article  ADS  Google Scholar 

  51. Mazzola, L., Laine, E.M., Breuer, H.P., Maniscalco, S., Piilo, J.: Phenomenological memory-kernel master equations and time-dependent Markovian processes. Phys. Rev. A 81, 062120 (2010)

    Article  ADS  Google Scholar 

  52. Vacchini, B.: A classical appraisal of quantum definitions of non-Markovian dynamics. J. Phys. B 45, 154007 (2012)

    Article  ADS  Google Scholar 

  53. Laine, E.M., Piilo, J., Breuer, H.P.: Measure for the non-Markovianity of quantum processes. Phys. Rev. A 81, 062115 (2010)

    Article  ADS  Google Scholar 

  54. Rivas, Á., Huelga, S.F., Plenio, M.B.: Entanglement and non-Markovianity of quantum evolutions. Phys. Rev. Lett. 105, 050403 (2010)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants Nos. 11274043, 11375025).

Author information

Authors and Affiliations

  1. School of Physics, Beijing Institute of Technology, Beijing, 100081, China

    Chao-Quan Wang, Jian Zou & Bin Shao

Authors
  1. Chao-Quan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jian Zou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bin Shao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chao-Quan Wang.

Appendices: The methods of non-Markovianity witness with or without initial correlation

Appendices: The methods of non-Markovianity witness with or without initial correlation

1.1 Appendix 1: Non-Markovianity witness without initial correlations

We introduce the usual degree of non-Markovianity (N) proposed in Refs. [47,48,49,50,51],

$$\begin{aligned} \textit{N}=\max _{\{\rho ^{N}_s(0), \rho _s^{N\perp }(0)\}}\int _{\Omega _+^N}\partial _t D(\rho ^{N}_s(t), \rho _s^{N\perp }(t))\mathrm{d}t, \end{aligned}$$
(42)

where \(D(\rho ^{N}_s(t), \rho _s^{N\perp }(t))=\frac{1}{2}\Vert \rho ^{N}_s(t)-\rho _s^{N\perp }(t)\Vert _1\) with \(\Vert .\Vert _1\) being the trace norm, is the trace distance between two evolved states and \(\Omega _+^N=\bigcup _i(a_i^N,b_i^N)\) is the union of all the time intervals \((a_i^N,b_i^N)\) in our observation window within which \(\partial _t D(\rho ^{N}_s(t), \rho _s^{N\perp }(t))>0\). \(\rho ^{N}_s(0)\) and \(\rho _s^{N\perp }(0)\) are two initial orthogonal states of the system, and their corresponding time-evolved states are \(\rho ^{N}_s(t)\) and \(\rho _s^{N\perp }(t)\). The maximization is performed over all possible orthogonal pairs of initial system states. As the evolution under scrutiny in this paper proceeds in discrete temporal steps, we will employ the discretized version of Eq. (42), which is obtained as [37, 52, 53]

$$\begin{aligned} \textit{N}= & {} \max _{\{\rho ^{n1}_s(0), \rho _s^{n1\perp }(0)\}}\sum _{n\in \Omega _+}[D(\rho ^{n1}_s(n+1), \rho _s^{n1\perp }(n+1))\nonumber \\&\quad -\,D(\rho ^{n1}_s(n), \rho _s^{n1\perp }(n))] \end{aligned}$$
(43)

with the time-evolved states of the system \(\rho ^{n1}_s(n)\) and \(\rho _s^{n1\perp }(n)\) obtained starting from a pair of the initial orthogonal states \(\rho ^{n1}_s(0)\) and \(\rho _s^{n1\perp }(0)\) after n steps of our protocol. \(\Omega _+=\bigcup _n(n,n+1)\) is the union of all the collision step intervals \((n,n+1)\) in our observation window within which \(D(\rho ^{n1}_s(n+1), \rho _s^{n1\perp }(n+1))-D(\rho ^{n1}_s(n), \rho _s^{n1\perp }(n))>0\).

1.2 Appendix 2: Non-Markovianity witness with initial quantum correlation

We consider the method which can be used to witness the non-Markovian dynamics when the initial correlation between the system and environment is present. Generally, the measure for non-Markovian dynamics is defined as namely the degree of non-Markovianity based on trace distance between different system states or the concurrence between system and reference system [47, 48, 54]. However, the two ways defining the degree of non-Markovianity both require that all the system states over the Bloch sphere have to be taken, and the state of environment must be the same, which indicates that the definition of the degree of non-Markovianity only is suitable when the system initially has no correlation with environment. So far, how to measure the non-Markovian dynamics with initial system–environment correlations is still an open question. In view of this, inspired by the previous definition of the degree of non-Markovianity, we make use of the non-monotonicity of the trace distance D[\(\rho ^{q}_{s}(n),\widetilde{\rho }^{q}_{s}(n)\)] between two states of system to witness the non-Markovian dynamics of an open quantum system with initial quantum correlation between the system and environment, and define

$$\begin{aligned} \mathrm{NE}^q=\sum _{n\in \Omega _+}\Delta D(n) , \end{aligned}$$
(44)

as the non-Markovian effect (NE) in this paper. Here \(\Delta D(n)=D[\rho ^{q}_{s}(n+1),\widetilde{\rho }^{q}_{s}(n+1)] -D[\rho ^{q}_{s}(n),\widetilde{\rho }^{q}_{s}(n)]\) with \(\widetilde{\rho }^{q}_{s}(n)\) being the state of system after 2n collisions with E-S corresponding to the initial state \(\widetilde{\rho }^{q}_{s,b}(0)=\mathrm{Tr}_b(\rho ^{q}_{s,b}(0))\otimes \mathrm{Tr}_s(\rho ^{q}_{s,b}(0))\) and here the definition of \(\Omega _+\) is the same as that in Eq. (43), in which \(\Delta D(n)>0\).

1.3 Appendix 3: Non-Markovianity witness with initial classical correlation

We introduce the method which makes use of the non-monotonicity of the trace distance between two states of system to witness the effect of initial classical correlation on the non-Markovian dynamics of the system. However, the quantification of the NE is expressed by an alternative method differently from Eq. (44), which can be written as

$$\begin{aligned} \mathrm{NE}^{ci}= & {} \max _{\{\delta \in [0,\pi ],\varphi \in [0,2\pi ]\}}\sum _{n\in \Omega _+}\Delta D(n)\end{aligned}$$
(45a)
$$\begin{aligned} \Delta D(n)= & {} D\left[ \rho ^{ci}_{s}(n+1),\widetilde{\rho }^{ci}_{s}(n+1)\right] -D\left[ \rho ^{ci}_{s}(n),\widetilde{\rho }^{ci}_{s}(n)\right] , \end{aligned}$$
(45b)

where \(i=1,2\) correspond respectively to collision model case I and collision model case II, \(\widetilde{\rho }^{ci}_{s}(n)\) is the system state after 2n collisions with E-S corresponding to the initial state \(\widetilde{\rho }^{ci}_{s,b}(0)=\widetilde{\rho }^{ci}_{s}(0)\otimes \mathrm{Tr}_s(\rho ^{ci}_{s,b}(0))\) with the initial system state \(\widetilde{\rho }^{ci}_{s}(0)=\cos \frac{\delta }{2}|2\rangle +\sin \frac{\delta }{2}e^{i\varphi }|0\rangle , \delta \in [0,\pi ],\varphi \in [0,2\pi ]\). The maximization is performed by taking all possible system states \(\widetilde{\rho }^{ci}_{s}(0)\) over the Bloch sphere. It is noted that the above definition is a little different from Eq. (44) where for the initial state \(\widetilde{\rho }^{q}_{s,b}(0)=\mathrm{Tr}_b(\rho ^{q}_{s,b}(0))\otimes \mathrm{Tr}_s(\rho ^{q}_{s,b}(0))\) the initial states of both the system and E-B are fixed, while here only the initial state of E-B is fixed and the same as Eq. (44), but the initial states of the system is taken over the whole Bloch sphere.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, CQ., Zou, J. & Shao, B. Analysis of various factors affecting the non-Markovian dynamics associated with a hierarchical environment based on collision model. Quantum Inf Process 16, 156 (2017). https://doi.org/10.1007/s11128-017-1604-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-017-1604-0

Keywords

Search

Navigation